Spike-adding in neuron systems induced by bifurcation or non-bifurcation: A comparative study via solution continuation

The phenomenon of spike-adding in biological neuron pacemakers and neuron models has been observed, which is crucial for understanding the information encoding of neurons. While there have been studies on the spike-adding mechanism driven by bifurcations, the generation mechanism of the first stable spike has not been fully revealed, and whether spike-adding under external stimulation necessarily relies on a bifurcation mechanism that leads to system instability remains a research gap. To deeply explore the potential diversity of spike-adding mechanisms in neurons, this paper conducts a systematic comparative study on the spike-adding phenomenon in the Hindmarsh-Rose (HR) model and the forced piezoelectric neuron model through the multiple shooting method and numerical continuation technique, combined with bifurcation diagrams, phase diagrams, time series diagrams, and Floquet theory. The results show that in the HR model, the generation of the first stable spike follows a new, bifurcation-driven path, which is produced from an unstable orbit through a subcritical inverse period-doubling bifurcation. The subsequent spike-adding processes are formed through saddle-node bifurcations and subcritical inverse period-doubling bifurcations. In contrast, in the forced piezoelectric neuron model, a non-bifurcation mechanism is presented, where the number of spikes can gradually increase by reducing the external excitation frequency, while maintaining the stability of the system. This paper systematically clarifies two different spike-adding mechanisms, which are of great significance for understanding the robustness and flexibility of neural information processing, providing a theoretical basis for the design of neural regulation strategies.